Download Treating some solid state problems with the Dirac equation

J. Phys. A: Math. Gen. 33 (2000) L509–L514. Printed in the UK
PII: S0305-4470(00)16417-0
LETTER TO THE EDITOR
Treating some solid state problems with the Dirac equation
R Renan, M H Pacheco and C A S Almeida
Universidade Federal do Ceará, Physics Department, CP 6030, 60470-455, Fortaleza-Ce, Brazil
E-mail: [email protected] and [email protected]
Received 15 August 2000
Abstract. The ambiguity involved in the definition of effective-mass Hamiltonians for
nonrelativistic models is resolved using the Dirac equation. The multistep approximation is
extended for relativistic cases allowing the treatment of arbitrary potential and effective-mass
profiles without ordering problems. On the other hand, if the Schrödinger equation is used,
our relativistic approach demonstrates that the two results are coincident if the BenDaniel–Duke
prescription for the kinetic-energy operator is implemented. Applications for semiconductor
heterostructures are discussed.
The effective-mass theory has been successfully used in semiconductor heterostructures [1].
An interesting aspect arises when we treat materials whose properties change from region to
region. In particular, when the effective mass depends on position, the Schrödinger equation
for an arbitrary potential profile is usually solved numerically by different methods. However,
one of the problems of the effective-mass theory for semiconductor heterostructures is to
decide how to write out the Hamiltonian operator. This problem arises from the canonical
quantization of the classical Hamiltonian. For position-dependent carrier effective mass, we
have an ordering problem with the kinetic-energy operator (KEO). Some authors proposed
different forms for the KEO, all having the generic form proposed by von Roos [2]
T̂ = 41 (mα (x)p̂mβ p̂mγ (x) + mγ (x)p̂mβ p̂mα (x))
(1)
where α + β + γ = −1, but the problem was not resolved because there is no first principle to
fix only one operator.
This ambiguity indicates that the Schrödinger equation is not rigorously suitable in the
effective-mass aproximation with position-dependent effective mass. It is reasonable to try
another equation that represents the same physics as the Schrödinger equation in the low-energy
limit.
The object of this Letter is to demonstrate that, in fact, the Dirac equation (at adequate
limits) can successfully be used to describe quantum mechanical systems where positiondependent effective mass is present. We recall that, in the Dirac equation, the KEO and the
mass term appear separately, so there is no ordering problem in this context.
Obviously the considerations of this Letter only concern the mathematical issues related
to the equations of motion. Indeed, a physically sensible application of the Dirac equation to
semiconductor heterostructures would have to take in to account a relativistic extension of the
Wannier–Slater theorem [3].
In this Letter we use a numerical method (multistep potential approximation [4]) which
has been applied to solve the Schrödinger equation for an arbitrary potential profile. Here
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Letter to the Editor
we extend this algorithm to the relativistic case, in such a way that the ambiguity problem
is overcome. In particular, we consider the Dirac equation with a one-dimensional arbitrary
potential well and find the energy levels for a particle. Also we apply the method for a
particular type of heterostructure and compare the results to those obtained in the context of
the Schrödinger equation with the form (1) for the KEO and several choices for the parameters
α, β and γ . For a KEO in the Schrödinger equation of the form suggested by BenDaniel and
Duke (β = −1, γ = 0) [5], we conclude that the two equations lead to the same solutions in
the energy range concerned.
Let us now introduce the numerical method that allows us to obtain the energy levels for
a Dirac equation with space-dependent effective mass.
We will consider a particle with mass m(z) that is subjected to an arbitrary one-dimensional
potential well V (z). The time-independent Dirac equation is written as (in units with
h̄ = c = 1) [8]
(α p̂ + βm(z)) = (E − V (z))
(2)
d
where p̂ = −i dz
is the momentum operator, E is the electron energy, α and β are 4 × 4
matrices given by
0 σ3
I
0
α=
β
=
(3)
σ3 0
0 −I
where I is the 2 × 2 identity matrix and σ 3 is a 2 × 2 Pauli matrix defined as
1 0
3
σ =
.
0 −1
(4)
Consider now an arbitrary well as sketched in figure 1. We split the interval [a, b] into
N infinitesimal intervals of length z = (b − a)/N . For the ith interval, we approximate the
potential and the mass by
V (z) = V (zi ) = Vi
and
m(z) = m(zi ) = mi
for
zi z < zi+1 .
(5)
The wavefunction of the electron in Dirac’s equation with no spin flip in the ith interval is




1
1
 0 
 0 
 + Bi e−ipi z  −p

pi
ψi (z) = Ai eipi z 
(6)
i
 E−v +m 
 E−v +m

i
i
0
i
i
0
where pi = (E − Vi )2 − m2i . The bound state conditions are given by |E − V0 | < m0 and
|E − VN +1 | < mN +1 . By imposing the continuity of the wavefunction at each z = zi , we have
a matrix M(E) which relates the coefficients in the region where z < a with the region where
z>b
AN +1
A0
= M(E)
.
(7)
BN +1
B0
The finiteness of the wavefunction requires that
M(E)22 = 0.
(8)
So, the solution of (8) gives us the energy levels.
Note that this numerical method is especially convenient for treating wells and barriers
with arbitrary profiles and it is nothing other than an extension of the transfer-matrix method
for relativistic theories. To the best of our knowledge, this is the first numerical analysis for
evaluating energy levels in an relativistic equation with mass position-dependent and arbitrary
Letter to the Editor
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a
zi
b
z
Figure 1. Generic profile of a one-dimensional quantum potential well. We split the interval [a, b]
into N infinitesimal intervals of length z = (b − a)/N .
V(z)
6
Vmax
?
z
m(z)
6
mmax
?
z
Figure 2. Potential well and effective mass, as a function of the
position, for an abrupt heterostructure.
potential. However, our main point here is to demonstrate that the Dirac equation can be
used to obtain unambiguously results in situations where the Schrödinger equation depends
on ordering problems.
As an illustration, we applied the method described above for an electron in a onedimensional GaAs/Al0.3 Ga0.7 As heterostructure. For the sake of comparison with previous
results we take a square well, as sketched in figure 2. The electron effective mass is 0.67m0
and 0.86m0 for GaAs and Al0.3 Ga0.7 As respectively (m0 is the free electron mass) (figure 2).
In the Schrödinger equation we use the von Ross operator [2] considering several values of the
α parameter. Note that for abrupt heterojunctions only Hamiltonians with α = γ are viable,
due to continuity conditions across the heterojunction [9]. As we can see in figure 3 there is an
extraordinary coincidence between the results from the Dirac and Schrödinger equations for
α = 0. As a matter of fact, this is expected since the maximum value of the energy involved
is 3 eV. Further, this result strongly supports the prescription of BenDaniel and Duke [5] for
the Schrödinger context.
As a second illustration consider the conduction-band structure of a GaAs/Al0.3 Ga0.7 As
system with nonabrupt interface and assume that the effective mass changes linearly at the
transition regions while the potential well varies quadratically in those regions (denoted by a)
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Letter to the Editor
ENERGY (meV)
250
Dirac
α= 0.0
α=0.5
α=−0.5
200
150
100
50
10
20
30
40
WELL WIDTH (Å)
50
60
Figure 3. Eigenenergies in the conduction band of GaAs/Al0.3 Ga0.7 As, with a conduction band
offset of 0.6 versus the well width (abrupt heterojunction). The chosen value for a is 20% of the
well width. The solid curve shows the calculations performed using the Dirac equation. The broken
curve shows the calculations for the Schrödinger equation using the BenDaniel–Duke prescription
(α = 0). The ++ curve denotes calculations using the Zhu–Kroemer prescription (α = −0.5).
6
V(z)
a
a
Vmax
?
6
z
a
m(z)
a
mmax
?
z
Figure 4. Potential well and effective mass, as a function of the
position, for a nonabrupt heterostructure (a denotes the transition
region).
as shown in figure 4. There the potential is given by
V (z) = C["1 χ (z) + "2 χ (z)2 ]
(9)
where C = 0.6 is the conduction band offset and "1 = 0.3, "2 = 0.7 are constants associated
with the compositional dependence of the energy-gap difference between GaAs and AlGaAs
(experimental parameters and details concerned can be seen in [6, 7]).
Letter to the Editor
L513
ENERGY (meV)
250
Dirac
α = 0.5
α= −0.5
α = 0.0
200
150
100
50
10
20
30
40
WELL WIDTH (Å)
50
60
Figure 5. Eigenenergies in the conduction band of GaAs/Al0.3 Ga0.7 As, with a conduction bandoffset of 0.6 versus the well width (nonabrupt heterojunction). The chosen value for a is 20% of
the well width. The solid curve shows the calculations performed using the Dirac equation and
the broken curve shows the calculations for the Schrödinger equation using the BenDaniel–Duke
prescription (α = γ = 0). The ++ curve denotes calculations using the Zhu–Kroemer prescription
(α = γ = −0.5).
Again, we use our relativistic method and the Schrödinger equation with the BenDaniel–
Duke prescription (α = γ = 0). Once more, as we can see in figure 5, a complete coincidence
between the relativistic and non-relativistic results is obtained.
In conclusion, we have shown that a relativistic method can be successfully used to
overcome the ordering problem of the KEO in non-relativistic models. Since the range of
energy involved is extremely low (comparing to electron rest mass), the numerical results are
perfectly coincident in both cases.
It is worthwhile to mention that, notwithstanding the Wannier–Slater theorem commented
on in the introduction, we claim attention for the coincidence shown above. Therefore,
we believe that, after exhaustive and appropriate considerations of the effective-mass
approximation, band structures and the periodic potential, the relativistic approach constructed
here could be used to calculate physical parameters in the theory of abrupt and nonabrupt
semiconductor heterostructures.
Moreover, the relativistic treatment developed here can be applied to all physical systems
described by a Sturm–Liouville eigenvalue equation (within an appropriate range of energy),
namely
d
1 df
−
+ q(z)f (z) = λw(z)f (z).
(10)
dz p(z) dz
Here f is an eigenfunction, λ is an eigenvalue and p, q and w describe particular properties of
the system. For example, this equation can describe the motion of electrons or phonons along
the growth axis of a [100] zinc-blende heterostructure [10]. Thereby, the equation (10) can
always be replaced by a corresponding Dirac equation to avoid ordering problems.
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Letter to the Editor
CASA and MHP were supported in part by Conselho Nacional de Desenvolvimento Cientı́fico
e Tecnológico-CNPq and Fundação Cearense de Amparo à Pesquisa-FUNCAP.
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